A scatter search algorithm for solving vehicle routing problem with loading cost

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<ul><li><p>hi</p><p>versi</p><p>m (Vitsg cn opher</p><p>vehicle routes. Considering the features of the VRPLC, a scatter search (SS) is proposed. By introducing</p><p>an imer (195e optimgeogr</p><p>vehicle routing problem (PVRP) that considers the service time asa period instead of a day (Francis &amp; Smilowitz, 2006); (5) splitdelivery vehicle routing problem (SDVRP) in which customer canbe served more than once (Belenguer, Martinez, &amp; Mota, 2000)and so on. The above-mentioned problems are special classes ofgeneral VRP and they correspond to some real scenarios of trans-</p><p>stant and neglected in the objective function. The loading cost isan important part of charge for consumption of gas that changesdue to the amount of the load of the vehicle. However, in real-world distribution activities, the vehicle load varies greatly fromone customer to another on a whole route, because the goods willbe unloaded when a vehicle visited the customer. The vehicle loadwill decrease as the vehicle visits customers one by one and nallyequals the quantity of demand at the last customer on a trip. Inpractice, when transporting some special freight, such as hazard-ous materials, perishable food, livestock or something with special</p><p>* Corresponding author. Tel.: +86 24 83673810.E-mail addresses: jftang@mail.neu.edu.cn (J. Tang), water_zj2000@yahoo.com.cn</p><p>Expert Systems with Applications 37 (2010) 40734083</p><p>Contents lists availab</p><p>w</p><p>.e(J. Zhang), panzhendong@ise.neu.edu.cn (Z. Pan).tomers to minimize the total operation cost. Since then, VRP hasreceived much attention with respects to various aspects of theproblem. They are: (1) capacitated vehicle routing problem (CVRP)that is an elementary version of VRP and only considers capacityconstraint (Laporte &amp; Nobert, 1983); (2) vehicle routing problemwith time windows (VRPTW) that considers the due service timebound of every customer (Bent &amp; Van Hentenryck, 2004); (3) mul-ti-depot vehicle routing problem (MDVRP) that considers multipledepots instead of a single depot (Lim &amp; Wang, 2005); (4) periodic</p><p>In general VRP models and its variants, the transportation costusually includes two parts: xed cost and variable cost. The xedpart is the dispatching cost for a vehicle, which depends only onthe volume of goods loaded in the vehicle and distance of a trip.The variable cost is in proportion to the sum of the distance trav-eled. In summary of the literature, the aforementioned modelsfor various forms of VRP consider the vehicle load as constant ona whole route, and hence the cost associated with the amount ofthe load, referred to loading cost hereafter in this paper, is a con-1. Introduction</p><p>Vehicle routing problem (VRP) islem proposed by Dantzig and Ramsned as the problem of designing thhomogeneous vehicles to serve some0957-4174/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.eswa.2009.11.027customer-oriented three-dimensional encoding method, sweep algorithm and optimal splitting proce-dure are combined to obtain better trial solutions. The arc combination and improved nearest neighborheuristic are adopted as a solution combination method and an improvement method to generate andimprove new solutions, respectively. Computational experiments were carried out on benchmark prob-lems of capacitated VRP with seven categories of distribution scenarios. The computational results showthat the SS is competitive and superior to other algorithms on most instances, and that the VRPLC canmore reasonably and exactly formulate the vehicle routing problem with more cost savings than generalVRP models.</p><p> 2009 Elsevier Ltd. All rights reserved.</p><p>portant scientic prob-9) rstly. It can be de-al routes for a eet of</p><p>aphically scattered cus-</p><p>portation in distribution management. Most of the aforementionedVRPs are formulated as 0-1 integer programming problems (Chu,2005; Lim &amp; Wang, 2005; Lu &amp; Dessouky, 2004; Mingozzi, Giorgi,&amp; Baldacci, 1999), 0-1 mixed integer programming problems(Bookbinder &amp; Reece, 1988; Tavakkoli, Safaei, Kah, &amp; Rabbani,2007) and network optimization models (Yi &amp; Ozdamar, 2007).loading cost may lead to sub-optimal routes. In this paper, we investigate VRP with loading cost (VRPLC),which considers the costs associated with the amount of the load on the vehicle when determining theA scatter search algorithm for solving ve</p><p>Jiafu Tang *, Jun Zhang, Zhendong PanKey Laboratory of Integrated Automation of Process Industry of MOE, Northeastern Uni</p><p>a r t i c l e i n f o</p><p>Keywords:Vehicle routing problemMeta-heuristic algorithmScatter searchLoading cost</p><p>a b s t r a c t</p><p>The vehicle routing problement. In classical VRP andtation; therefore the loadinthe objective function whefrom one customer to anot</p><p>Expert Systems</p><p>journal homepage: wwwll rights reserved.cle routing problem with loading cost</p><p>ty, Shenyang, PR China</p><p>RP) is an important scientic problem addressed in distribution manage-variants, the vehicle load is often regarded as a constant during transpor-ost associated with the amount of the load on the vehicle, are neglected intimizing a vehicle routine. However, in real-world, the vehicle load variesin a vehicle route. Thus, the vehicle route without considering the effect of</p><p>le at ScienceDirect</p><p>ith Applications</p><p>lsevier .com/locate /eswa</p></li><li><p>ented three-dimensional encoding and combines the sweep and</p><p>h Aphandling, extra cost can be charged in real-world distributionactivity. This part of cost is commonly a function of weight of theload and accounts for about fty percent of the transportation cost.</p><p>In fact, the vehicle load is considered as a kind of variable, calledow variable, to formulate traveling salesman problem and a vari-ety of related transportation problems, e.g. network ow problem(Dumas, Desrosiers, &amp; Soumis, 1991; Gavish &amp; Graves, 1978; Gou-veia, 1995). The load variable appears in constraints or appears as athreshold in objective function. Thus, the aforementioned VRPmodels without considering the loading cost could only providean approximate and simplied formulation of practical VRP, andsubsequently the vehicle routes determined without consideringthe effect of loading cost may lead to inexact optimal routes.</p><p>This paper focuses on a new formulation of VRP, short for VRPLChereafter in the paper, taking into account the loading cost in theobjective function when optimizing vehicle routes. Distinguishedfrom the traditional models of VRP, the loading cost is regardedas a variable in objective function rather than zero. When the load-ing cost is zero, the VRPLC is CVRP.</p><p>CVRP is a NP-hard problem and has been intensively studied. Asa generalization of CVRP, VRPLC is a NP-hard problem. There hasbeen no report on solution algorithms for VRPLC in the literature.However, algorithms for solving various CVRP receivedmuch atten-tion. The algorithms for solving CVRP can be divided into two cate-gories: exact algorithms and heuristics. The detailed survey of exactalgorithms for CVRP can be found in Cordeau, Laporte, Savelsbergh,and Vigo (2007). The largest CVRP instance that has been solvedoptimally using an exact algorithm is of 134 customers (Fukasawaet al., 2006). With the size of instances increasing, the computationtime of exact algorithm becomes intolerable. Thus, heuristic algo-rithms have received more attention and they are suitable to solvepractical instances. The heuristics for solving CVRP and its variantsare classied into two categories: classical heuristic and meta-heu-ristic. The famous classical heuristics are those like CW saving heu-ristic (Clarke &amp; Wright, 1964), sweep algorithm (Gillett &amp; Miller,1974), insertion heuristic (Gendreau, Hertz, Laporte, &amp; Stan, 1998)and other iterative improvement heuristics (Tan, Lee, Zhu, &amp; Ou,2001; Waters, 1987). The classical heuristic algorithms terminatewith satisfactory solution quickly and are easy to be implemented;but, in general, the gap between the heuristic solution and the opti-mal solution is large. A systematic review on classical heuristics canbe found in the literatures (Laporte, Gendreau, Potvin, &amp; Semet,2000; Laporte, 2007; Laporte &amp; Semet, 2002).</p><p>Compared with classical heuristic, meta-heuristic algorithmscan obtain better solutions than classical heuristics or even globaloptimal solutions in a reasonable time. The well-known meta-heu-ristic algorithms include tabu search (TS) (Glover, 1986), simulatedannealing (SA) (Kirkpatrick &amp; Gelatt, 1983), genetic algorithm(GA) (Mitchell, 1996), ant colony optimization algorithm (ACO)(Doerner, Hartl, Benkner, &amp; Lucka, 2006) and neural networks(Smith, 1999). TS and SA are based on local search principle (Kytj-oki, Nuortio, Brysy, &amp; Gendreau, 2007) and usually start searchingfrom one initial solution and explore more promising solutions inthe solution region. Some efcient implementation of TS and SAfor CVRP can be found in Tavakkoli-Moghaddam et al. (2007) andBrando (2009). GA and ACO are representative methods ground-ing in population search principle (Smith, 1999). The meta-heuris-tics based on population search principle maintain a pool ofsolutions and update the solutions in the pool by rules. Recently,several ACO algorithms are developed to solve CVRP successfully(Doerner et al., 2006). Over the last 2 year, some meta-heuristicalgorithms, like variable neighborhood search (VNS) (Kytjokiet al., 2007), large neighborhood search (LNS) (Goel &amp; Gruhn,</p><p>4074 J. Tang et al. / Expert Systems wit2008), adaptive large neighborhood search(ALNS) (Pisinger &amp;Ropke, 2007), scatter search (SS) (Russell &amp; Chiang, 2006) are ap-plied to the vehicle routing problem. Gendreau, Laporte, and Potvinoptimal splitting procedure (Prins, 2004) to construct the initialtrial solutions pool. New solutions are generated by combiningthe arc selected. The improved nearest neighbor method is usedto improve new solutions. Seven types of benchmark datasets ofCVRP with different node distribution are chosen, in the form ofrandom, cluster (one and several), mixing random and cluster, reg-ulation(dispersive and dense), and similar magnetic eld. Theexperiments to test the performance, stability and sensitivity ofparameters of the SS on benchmark problems are conducted.</p><p>Following the introduction, the paper is organized as follows.The mathematical model is described in Section 2, along with theproblem assumption. In Section 3, combining the features of themodel, a SS is developed to solve the VRPLC. In Section 4, compu-tational results of the SS on seven types of benchmark instancesare reported. Finally, conclusions are made in Section 5.</p><p>2. Mathematical model for VRPLC</p><p>Consider a distribution network in which one product isshipped from a depot to a set of customers. The VRPLC can be de-ned on a graph G = (V,A), where V is vertex set and A is the arc set.The vertex set V includes the depot v0 and customers Vc fv1;v2; . . . ;vNg. The index of the depot is 0 while customers are in-dexed from 1 to N. A fv i; v jjv i; v j 2 V ; i jg is the arc set. Adistance matrix D dij is dened on A, which is associated toeach arc v i;v j. We assume that D is symmetric and satises thetriangle inequality. The depot owns homogeneous vehicles, andthe vehicle number is assumed to be unlimited. Each customersdemand is known and less than the vehicle capacity. The followingnotations are used throughout the paper to formulate the model:</p><p>N = the number of the customers;K = the number of the vehicles;Q = the capacity of the vehicle;Cd = the cost of traveling per unit-distance by a vehicle, calleddistance coefcient hereafter;Cg = the cost of delivering product of per unit-weight and perunit-distance, called load coefcient hereafter;Cv = the xed cost of dispatching a vehicle called vehicle coef-(2002) Cordeau, Gendreau, Hertz, Laporte, and Sormany (2005)gave an extensive review of the above-mentioned meta-heuristicsin the past decades and recent years.</p><p>Among the aforementioned meta-heuristics, scatter search (SS)can be regarded as an efcient algorithms for VRP (Glover, Laguna,&amp; Marti, 2000). SS is a population-based meta-heuristic and intro-duced by Glover (1977) rstly for solving integer programming.Distinguished from other population-based meta-heuristic, SSoperates on some solutions called reference set (Glover, Laguna, &amp;Marti, 2003). It selects more than two solutions of reference set ina systematic way aiming at producing new solutions. In the domainof solving routing problem the SS can yield better results than someoften used meta-heuristics (Alegre, Laguna, &amp; Pacheco, 2007; Mota,Campos, &amp; Corbern, 2007; Russell &amp; Chiang, 2006).</p><p>In this paper, we study VRP with loading cost (VRPLC), whichdetermines a route in order to minimize the total transportationcost considering the costs incurred by vehicle load in objectivefunction. With the formulation of VRPLC, the vehicle load is viewedas varying from a customer to another one instead of a constantduring a trip. Thus, the loading cost brought by the volumes ofgoods loaded in a vehicle is considered as a part of the transporta-tion cost. Considering the features of VRPLC, a scatter search (SS)algorithm is developed. The SS algorithm adapts customer-ori-</p><p>plications 37 (2010) 40734083cient hereafter;dij = the distance from node i to j, i, j = 0,1,2, . . . ,N;</p></li><li><p>k = 0,1,2, . . . ,K.</p><p>and Chiang (2006) solved a classical VRPTW by SS. Their SS alsoadopts two different construction heuristics to generate the initialtrial solutions. A common arc method and an optimization-based</p><p>h Applications 37 (2010) 40734083 4075The VRPLC aims to minimize the transportation cost includingdistance cost, loading cost brought by the weight of goods distrib-uted and the dispatching cost. It is formulated as follows:</p><p>Min costXN</p><p>i0</p><p>XN</p><p>j0dijXK</p><p>k1xijkCdCgyijkCv</p><p>XN</p><p>j1</p><p>XK</p><p>k1x0jk 1</p><p>Subject toXN</p><p>i1qiXN</p><p>j0xijk6Q ; k2f1;2; . . . ;Kg 2</p><p>XN</p><p>j0</p><p>XK</p><p>k1xijk 1; i2f1;2; . . . ;Ng 3</p><p>XN</p><p>i0</p><p>XK</p><p>k1xijk 1; j2f1;2; . . . ;Ng 4</p><p>XN</p><p>j1x0jk61; k2f1;2; . . . ;Kg 5</p><p>XN</p><p>j1x0jkXN</p><p>j1xj0k 0; k2f1;2; . . . ;Kg 6</p><p>X</p><p>i2S</p><p>X</p><p>j2Sxijk6 jSj1 8S#Vc; 26 jSj6n; 7</p><p>k2f1;2; . . . ;KgXN</p><p>j0ij</p><p>XK</p><p>k1xjikyjikxijkyijk qi; i2f1;2; . . . ;Ng 8</p><p>06 yijk6Qxijk; i; j2f0;1; . . . ;Ng;k2f1;2; . . . ;Kg 9xijk 2f0;1g; i; j2f0;1; . . . ;Ng; k2f1;2; . . . ;Kg 10</p><p>The objective function includes the distance cost, the loadingcost brought by the weight of freight and the dispatching cost.The constraint (2) is the vehicle capacity constraint. The formula(3)(5) represent that every customer is visited exactly once andthat served only by one vehicle. The constraint (6) guarantees thatthe vehicles starts from and return to the depot. The constraint (7)is the sub-tour elimination constraint. Eq. (8) shows the logic rela-tionship between the demand of customer i and the vehicle loadson the two arcs linking customer i.</p><p>The mathematical formulation of VRPLC can be regarded as thegeneralization of CVRP, since the model of VRPLC can be trans-formed to the classical CVRP when the Cg and Cv are equal to 0and neglecting the constraints (8) and (9). In the next section, ascatter search algorithm is proposed.</p><p>3. A scatter search algorithm to solve VRPLC</p><p>In recent years, mu...</p></li></ul>

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